Bob McClure created this web page, which is an excerpt from the 3 meg PDF file on how LaCoste and Romberg made their zero length tension springs.
The original document is http://www.gravitymeter-repair.com/images/gdmanual.pdf, the LaCoste and Romberg Instruction Manual for the model G & D Gravity Meters.  The idea of winding a spring and turning it "inside out" to make it a negative length spring was intriguing to Bob and is also to me!

PHYSICS OF THE GRAVITY SENSOR

In the early days of earthquake seismology, long period horizontal motions could be measured with the horizontal pendulum seismograph. As the axis of rotation became closer to vertical, the period became longer. Theoretically, if the axis is vertical, the period is infinite.

Dr. Romberg posed the question to his student, Lucien LaCoste, how to design a vertical seismograph with the characteristics as good as the existing horizontal pendulum seismograph.

In the illustrated suspension, there are two torques: gravitational and spring. If these two torques balance each other for any angle of the beam, the system will have infinite period. The smallest change in vertical acceleration (or gravity) will cause a large movement.

The torque due to gravity is:

Where W is the mass and d is the distance from the mass to the beam’s hinge.

The torque due to the spring is the product of the pull of the spring and the springs lever arm, s.

The length of the spring is r and by the law of sines:

If the spring constant is k and the length of the spring without force is n, The spring force is illustrated by this graph.

The torque due to the spring is then:

The total torque is:

This equation would yield zero torque and would be satisfied for all angles of theta if:

n = 0 and Wd - kab = 0

For n to equal zero, we must have a “zero length spring”. That is , a spring whose force-length graph passes through the origin or, at least, points toward the origin. The turns of a helical spring of zero unstressed length would bump into each other before the spring actually reached zero length. By making a helical spring whose turns press against each other when there is no force on the spring ,a “zero length spring” can be made.

ZERO LENGTH SPRING

There are several ways to make a zero length spring. A simple zero-length spring is a flat spiral spring. The mechanical properties of a spiral spring are not as convenient as a helical spring. To make a zero-length helical spring, the spring wire can be wound onto a mandrel. As the wire is wound, it can be twisted.

Another method (above) is to hold the wire at an angle and with tension while winding it on a rotating mandrel.

Still another method (above) is to “turn the spring inside out”.

The actual spring used in the L and R meters are “negative-length”. The spring wire is large enough and stiff enough that the spring would not act like an ideal spring if the spring were to be clamped at both ends. Thus, a very fine but strong wire is attached to the top end of the spring and another to the bottom of the spring. The top wire is clamped to the lever system and the bottom wire is clamped to the beam. The effective length of the spring is the combined length of the helical spring and the two fine wires. That combination is “zero-length”. The helical spring by itself is “negative-length”.

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